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A172088
Triangle: T(n,m) = n!! - m!! - (n-m)!! read by rows 0 <= m <= n, where ()!! are the double factorials.
1
-1, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, 4, 4, 4, -1, -1, 6, 10, 10, 6, -1, -1, 32, 38, 42, 38, 32, -1, -1, 56, 88, 94, 94, 88, 56, -1, -1, 278, 334, 366, 368, 366, 334, 278, -1, -1, 560, 838, 894, 922, 922, 894, 838, 560, -1, -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1
OFFSET
0,12
COMMENTS
Row sums are {-1, -2, -2, -2, 10, 30, 180, 474, 2322, 6426, 31536, ...}; n-th row sum is (n+1)*n!! - 2*A129981(n).
FORMULA
T(n,m) = A006882(n) - A006882(m) - A006882(n-m).
EXAMPLE
Triangle begins
-1;
-1, -1;
-1, 0, -1;
-1, 0, 0, -1;
-1, 4, 4, 4, -1;
-1, 6, 10, 10, 6, -1;
-1, 32, 38, 42, 38, 32, -1;
-1, 56, 88, 94, 94, 88, 56, -1;
-1, 278, 334, 366, 368, 366, 334, 278, -1;
-1, 560, 838, 894, 922, 922, 894, 838, 560, -1;
-1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1;
MAPLE
A172088 := proc(n, m)
doublefactorial(n)-doublefactorial(m)-doublefactorial(n-m) ;
end proc:
seq(seq(A172088(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Oct 11 2011
MATHEMATICA
T[n_, k_] = n!! -k!! -(n-k)!!; Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten
PROG
(PARI) f2(n) = prod(j=0, (n-1)\2, n-2*j);
T(n, k) = f2(n) - f2(k) - f2(n-k); \\ G. C. Greubel, Dec 05 2019
(Magma) F2:=func< n | &*[n..2 by -2] >;
[F2(n) - F2(k) - F2(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 05 2019
(Sage)
def T(n, k): return (n).multifactorial(2) - (k).multifactorial(2) - (n-k).multifactorial(2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 05 2019
CROSSREFS
Sequence in context: A243756 A172985 A117683 * A159891 A201941 A180060
KEYWORD
sign,tabl,easy
AUTHOR
Roger L. Bagula, Jan 25 2010
STATUS
approved